Start by identifying objectives (including scope) of the process in the context of your organization.

Or to put it another way, ask those who gave you the task to explain what they want the process to do. This will include defining what constitutes a problem in your organization.

The ITIL guidance relates to IT service management and this is quite a different activity from software development._________________"Method goes far to prevent trouble in business: for it makes the task easy, hinders confusion, saves abundance of time, and instructs those that have business depending, both what to do and what to hope."
William Penn 1644-1718

1 - Why you ? Why are you being asked to put in a PM process - which is sort of IT SM / ITIL not SDLC instead of the ITIL Certified folk. Why are they not the drivers for this and your team are involved. Does the company not understand what PM is

As ITSM / ITIL PM is finding the root cause of service impacting incidents - esepcially if the R.C is unknown as well as providing a solution, how does this fit with the S/W Team

Do you a) develop custom application and support the developed applications until the application has cease viability b) support the developed applicationa and develop bug fixes and releases c) something else